Journal of Applied MathematicsVolume 2012 (2012), Article ID 436531, 12 pagesdoi:10.1155/2012/436531

Research Article

Extended Precise Large Deviations of Random Sums in the Presence of END Structure and Consistent Variation

Shijie Wang¹ and Wensheng Wang²

¹School of Mathematic Sciences, Anhui University, Hefei, Anhui 230601, China²Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China

Received 6 November 2011; Accepted 30 December 2011

Academic Editor: Ying U. Hu

Copyright © 2012 Shijie Wang and Wensheng Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The study of precise large deviations of random sums is an important topic in insurance and finance. In this paper, extended precise large deviations of random sums in the presence of END structure and consistent variation are investigated. The obtained results extend those of Chen and Zhang (2007) and Chen et al. (2011). As an application, precise large deviations of the prospective- loss process of a quasirenewal model are considered.

1. Introduction

In the risk theory, heavy-tailed distributions are often used to model large claims. They play a key role in some fields such as insurance, financial mathematics, and queueing theory. We say that a distribution function belongs to the class if Such a distribution function is usually said to have a consistently varying tail. The heavy-tailed subclass was also studied by Cline and Samorodnitsky [1] who called it “intermediate regular variation.” Another well-known class is called the dominated variation class (denoted by ). A distribution function supported on is in if and only if for any (or equivalently for some ). For more details of other heavy-tailed subclasses (e.g., , and so on) and their relations, we refer the reader to [2] or [3].

Throughout this paper, let be a sequence of real-valued random variables with common distribution function and finite mean . Let be a nonnegative integer valued counting process independent of with mean function , which tends to infinity as . In insurance and finance, and always denote the claims and claim numbers respectively. Hence, randomly indexed sums (random sums), which denote the loss process of the insurer during the period , can be written as

Recently, for practical reasons, precise large deviations of random sums with heavy tails have received a remarkable amount of attention. The study of precise large deviations is mainly to describe the deviations of a random sequence or a stochastic process away from its mean. The mainstream research of precise large deviations of focuses on the study of the asymptotic relation which holds uniformly for some -region as . The study of precise large deviations of random sums was initiated by Klüppelberg and Mikosch [4], who presented several applications in insurance and finance. For some latest works, we refer the reader to [2, 3, 5–11], among others.

In this paper, we are interested in the deviations of random sums away from with any fixed real number . We aim at proving the following asymptotic relation: and the uniformity of (1.5). That is to say, in which -region as (1.5) holds uniformly. It is interesting that (1.5) reduces to (1.4) with replaced by . Hence, we call (1.5) the extended precise large-deviation probabilities. More interestingly, setting and replacing with in (1.5), where denotes the safety loading coefficient, now (1.5) reduces to precise large-deviation probabilities for prospective-loss process. About precise large deviations for prospective-loss process, we refer the reader to [5].

The basic assumption of this paper is that is extended negatively dependent (END). The END structure was firstly introduced by Liu [12].

Definition 1.1. One calls random variables END if there exists a constant such that hold for each , and all .

Recall that are called ND if both (1.6) and (1.7) hold with ; they are called positively dependent (PD) if inequalities (1.6) and (1.7) hold both in the reverse direction with . According to Liu’s [12] interpretation, an ND sequence must be an END sequence. On the other hand, for some PD sequences, it is possible to find a corresponding positive constant such that (1.6) and (1.7) hold. Therefore, the END structure is substantially more comprehensive than the ND structure in that it can reflect not only a negative dependence structure but also a positive one to some extent.

Under the assumption that is an ND sequence, Liu [6] and Chen and Zhang [7] investigated precise large deviations of random sums of nonnegative random variables and real-valued random variables, respectively. For a slightly more general dependence of END structure, Chen et al. [11] obtained precise large deviations of random sums of nonnegative random variables and random sums of real-valued random variables with mean zero centered by a constant . Up to now, to the best of our knowledge, little is known about extended precise large deviations of random sums in the presence of END structure and heavy tails. Our obtained results extend those of Chen and Zhang [7] and Chen et al. [11].

The rest of this paper is organized as follows. Section 2 gives some preliminaries. Precise large deviations of random sums in the presence of END real-valued random variables are presented in Section 3. In Section 4 we consider precise large deviations of the prospective-loss process of a quasirenewal model as an application of our main results.

2. Preliminaries

Throughout this paper, by convention, we denote . For two positive infinitesimals and satisfying we write if ; if ; if ; if ; if both and write if . For theoretical and practical reasons, we usually equip them with certain uniformity. For instance, for two positive bivariate functions and , we say that holds as uniformly for all in the sense that For a distribution, set where . In the terminology of Tang and Tsitsiashvili [13], is called the upper Matuszewska index of . Clearly, if , then . It holds for every that Moreover, if the distribution has a finite mean. See [11].

Next we will need some lemmas in the proof of our theorems. From Lemma 2.3 of Chen et al. [11] with a slight modification, we have the following lemma.

Lemma 2.1. Let be a sequence of real-valued END random variables with common distribution function . If , then, for every fixed and some , the inequality holds for all , and .

Lemma 2.2 below is a reformulation of Theorem 2.1 of [12], which is one of the main results in [12].

Lemma 2.2. Let be a sequence of real-valued END random variables with common distribution function and finite mean , satisfying Then, for any fixed , relation holds uniformly for all .

3. Main Results and Their Proofs

In this sequel, all limiting relationships, unless otherwise stated, are according to . To state the main results, we need the following two basic assumptions on the counting process .

Assumption 3.1. For any and some ,

Assumption 3.2. The relation holds for all .

Remark 3.3. One can easily see that Assumption 3.1 or Assumption 3.2 implies that See [5, 11].

Theorem 3.4. Let be a sequence of END real-valued random variables with common distribution function having finite mean and satisfying (2.6), and let be a nonnegative integer-valued counting process independent of satisfying Assumption 3.1. Let be a real number; then, for any , the relation (1.5) holds uniformly for .

Theorem 3.5. Let be a sequence of END real-valued random variables with common distribution function having finite mean and satisfying (2.6), and let be a nonnegative integer valued counting process independent of . (i)Assume that satisfies Assumption 3.1 and is a real number (regardless of or ), then for any fixed , the relation (1.5) holds uniformly for .(ii)Assume that satisfies Assumption 3.2 and is a negative real number; then, for any fixed , the relation (1.5) holds uniformly for .

Remark 3.6. One can easily see that Theorem 3.4 extends Theorem 3.1 of [11] with replaced by . On the other hand, replacing with , setting , and noticing that , (3.4) yields Theorem 4.1(i) of [11].

Remark 3.7. Under the conditions of Theorem 3.5, choosing , one can easily see that the relation (1.4) holds uniformly over the -region for arbitrarily fixed . Hence, Theorem 3.5 extends Theorem 1.2 of [7].

Proof of Theorem 3.4. We use the commonly used method with some modifications to prove Theorem 3.4. The starting point is the following standard decomposition: where we choose such that .We first deal with . Note that . Thus, as and uniformly for , it follows from Lemma 2.2 thatNext, for , noticing that , as and uniformly for , Lemma 2.2 yields that On the other hand, Finally, to deal with , we formulate the remaining proof into two parts according to and . In the case of , setting in Lemma 2.1 with , for sufficiently large and , there exists some constant such that In the case of , note that since . Similar to (3.8), for sufficiently large and , there exists some constant such that As a result, by (2.4), as and uniformly for , both (3.8) and (3.9) yield that where in the last step, we used Substituting (3.5), (3.6), (3.7), and (3.10) into (3.4), one can see that relation (1.5) holds by the condition and the arbitrariness of .

Proof of Theorem 3.5. (i) We also start with the decomposition (3.4).For and , note that since . Hence, mimicking the proof of Theorem 3.4, we obtain, as and uniformly for , where, in the last step, we used the relation Again, as and uniformly for ,Finally, in , setting in Lemma 2.1 with , by (2.4) and Assumption 3.1, as and uniformly for , there exists a constant such that where in the last step we also used (3.14). Combining (3.12), (3.15), and (3.16), relation (1.5) holds by the condition and the arbitrariness of .(ii) We also start with the representation (3.4) in which we choose such that .To deal with , arbitrarily choosing , we split the -region into two disjoint regions as For the first -region , noticing that , by Lemma 2.2, it holds uniformly for all that where the second step and the before the last before the step can be verified, respectively, as For the second -region , note that Hence, by Assumption 3.2, we still obtain uniformly for all . As a result, the relation holds uniformly for all .For , since , it holds that It follows from Lemma 2.2 that for all Symmetrically, Finally, in , note that . Therefore, Lemma 2.2 implies, as and uniformly for , thatSubstituting (3.22), (3.24), (3.25), and (3.26) into (3.4) and letting , the proof of (ii) is now completed.

4. Precise Large Deviations of the Prospective-Loss Process of a Quasirenewal Model

In this section we consider precise large deviations of the prospective-loss process of a quasirenewal model, where the quasi-renewal model was first introduced by Chen et al. [11]. It is a nonstandard renewal model in which innovations, modeled as real-valued random variables, are END and identically distributed, while their interarrival times are also END, identically distributed, and independent of the innovations.

Let be a sequence of END real-valued random variables with common distribution function and finite mean , satisfying (2.7). Let be a quasi-renewal process defined by where , independent of , form a sequence of END nonnegative random variables with common distribution nondegenerate at zero and finite mean . By Theorem 4.2 of [11], as , By Chen et al. [11], for any , , and some , where in the last step we use (4.10) in [11]. Thus, one can easily see that satisfies Assumption 3.1. Assume that also satisfies Assumption 3.2. Let be the safety loading coefficient. Replacing with and setting in Theorems 3.4 and 3.5, then, for any fixed , the relation holds uniformly for .

Acknowledgments

The authors would like to thank an anonymous referee for his/her constructive and insightful comments and suggestions that greatly improved the paper. This work was partially supported by NSFC Grant 11071076, the Talents Youth Fund of Anhui Province Universities (2011SQRL012ZD), the Project Sponsored by the Doctoral Scientific Research Foundation of Anhui University, and the 211 Project of Anhui University (2009QN020B).

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